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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A129092 a(n) = A030067(2^n - 1) for n >= 1, where A030067 is the semi-Fibonacci numbers.

Original entry on oeis.org

1, 2, 5, 16, 69, 430, 4137, 64436, 1676353, 74555322, 5777029421, 792086153688, 194591768192733, 86534148901444102, 70244955881077121873, 104827174339054175240700, 289320796542222620694103961
Offset: 1

Views

Author

Paul D. Hanna, Mar 29 2007

Keywords

Examples

			The semi-Fibonacci sequence (A030067) starts:
[(1), 1, (2), 1, 3, 2, (5), 1, 6, 3, 9, 2, 11, 5, (16), 1, ...],
and obeys the recurrence:
A030067(n) = A030067(n/2) when n is even; and
A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.
This sequence also equals row sums of triangle A129100:
    1;
    1,    1;
    2,    2,    1;
    5,    6,    4,   1;
   16,   24,   20,   8,   1;
   69,  136,  136,  72,  16,  1;
  430, 1162, 1360, 880, 272, 32, 1; ...
where columns of A129100 shift left under matrix square,
so that A129100^2 starts:
     1;
     2,    1;
     6,    4,   1;
    24,   20,   8,   1;
   136,  136,  72,  16,  1;
  1162, 1360, 880, 272, 32, 1; ...

Crossrefs

Cf. A030067, A129093, A129094.

Programs

  • PARI
    /* Generated as column 0 of triangle A129100: */ a(n)=local(A=Mat(1),B);for(m=1,n+1,B=matrix(m,m);for(r=1,m,for(c=1,r, if(r==c || r==1 || r==2,B[r,c]=1,if(c==1,B[r,1]=sum(i=1,r-1,A[r-1,i]), B[r,c]=(A^(2^(c-1)))[r-c+1,1])); )); A=B); return(A[n+1,1])

Formula

Equals the row sums and first column of triangle A129100: a(n) = A129100(n,0), where column 0 of matrix power A129100^(2^k) = column k of A129100 for k > 0.

A129101 Column 1 of triangle A129100; also equals column 0 of the matrix square of A129100.

Original entry on oeis.org

1, 2, 6, 24, 136, 1162, 15702, 346768, 12836904, 814033666, 90074891654, 17659668432744, 6211830230882472, 3960850942657072026, 4617438765658479411542, 9912250203901899238148640
Offset: 0

Views

Author

Paul D. Hanna, Mar 29 2007

Keywords

Crossrefs

Cf. A129100 (triangle); A129092 (column 0), A129102 (column 2), A129103 (column 3).

Programs

  • PARI
    a(n)=local(A=Mat(1),B);for(m=1,n+2,B=matrix(m,m);for(r=1,m,for(c=1,r, if(r==c || r==1 || r==2,B[r,c]=1,if(c==1,B[r,1]=sum(i=1,r-1,A[r-1,i]), B[r,c]=(A^(2^(c-1)))[r-c+1,1])); )); A=B); return(A[n+2,2])

Formula

a(n) = A129094(n+1) - A129093(n); a(n) = A030067(2^(n+1)+2^n-1) - A030067(2^(n+1)-3) for n>=0 where A030067 is the semi-Fibonacci numbers.

A129093 a(n) = A030067(2^(n+1) - 3) for n>=1, where A030067 is the semi-Fibonacci numbers.

Original entry on oeis.org

1, 3, 11, 53, 361, 3707, 60299, 1611917, 72878969, 5702474099, 786309124267, 193799682039045, 86339557133251369, 70158421732175677771, 104756929383173098118827, 289215969367883566518863261
Offset: 1

Views

Author

Paul D. Hanna, Mar 29 2007

Keywords

Examples

			This sequence also equals the row sums of the triangle formed from the semi-Fibonacci numbers (A030067) with 2^n terms in row n for n>=0:
n=0: 1;
n=1: 1, 2;
n=2: 1, 3, 2, 5;
n=3: 1, 6, 3, 9, 2, 11, 5, 16;
n=4: 1, 17, 6, 23, 3, 26, 9, 35, 2, 37, 11, 48, 5, 53, 16, 69; ...
and the rightmost border equals A129092(n) = A030067(2^n - 1).
The semi-Fibonacci numbers (A030067) start:
[1, (1), 2, 1, (3), 2, 5, 1, 6, 3, 9, 2, (11), 5, 16, 1, ...],
and obey the recurrence:
A030067(n) = A030067(n/2) when n is even; and
A030067(n) = A030067(n-1) + A030067(n-2) when n is odd.
This sequence also equals row sums of matrix square A129100^2:
1;
2, 1;
6, 4, 1;
24, 20, 8, 1;
136, 136, 72, 16, 1;
1162, 1360, 880, 272, 32, 1; ...

Crossrefs

Cf. A030067, A129092, A129094.

Programs

  • PARI
    /* As row sums of the matrix square of triangle A129100: */
a(n)=local(A=Mat(1),B);for(m=1,n+1,B=matrix(m,m);for(r=1,m,for(c=1,r, if(r==c || r==1 || r==2, B[r,c]=1, if(c==1, B[r,1]=sum(i=1,r-1,A[r-1,i]), B[r,c]=(A^(2^(c-1)))[r-c+1,1])))); A=B); sum(k=1,n,(A^2)[n,k])
for(n=1,20,print1(a(n),", "))

Formula

Equals the first differences of A129092: a(n) = A129092(n+1) - A129092(n).
Equals the row sums of the matrix square of triangle A129100.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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