A081495 Start with Pascal's triangle; form a rhombus by sliding down n steps from top on both sides then sliding down inwards to complete the rhombus and then deleting the inner numbers; a(n) = sum of entries on perimeter of rhombus.
1, 5, 17, 55, 189, 681, 2519, 9451, 35765, 136153, 520695, 1998745, 7696467, 29716025, 115000947, 445962899, 1732525861, 6741529113, 26270128535, 102501265057, 400411345659, 1565841089321, 6129331763923, 24014172955545, 94163002754699, 369507926510401
Offset: 1
Keywords
Examples
The rhombus pertaining to n = 4 is obtained from the solid rhombus .....1 ...1...1 .1...2...1 1..3...3...1 ..4..6...4 ...10..10 .....20 giving .....1 ...1...1 .1.......1 1..........1 ..4......4 ...10..10 .....20 and the sum of all the numbers is 55, a(4) = 55.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
Programs
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GAP
B:=Binomial;; Concatenation([1], List([2..25], n-> B(2*n, n)-B(2*(n-1), n-1) +2*n -3)); # G. C. Greubel, Aug 13 2019
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Magma
C:=Catalan; [1] cat [(n+1)*C(n) -n*C(n-1) +2*n-3: n in [2..25]]; // G. C. Greubel, Aug 13 2019
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Maple
seq(coeff(series(((1-x)^3 - (1-2*x-x^3)*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x) ), x, n+1), x, n), n = 1..25); # G. C. Greubel, Aug 13 2019
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Mathematica
With[{C = CatalanNumber}, Table[If[n==1, 1, (n+1)*C[n] -n*C[n-1] +2*n-3], {n, 25}]] (* G. C. Greubel, Aug 13 2019 *) -
PARI
vector(25, n, b=binomial; if(n==1,1,b(2*n, n)-b(2*(n-1), n-1) +2*n -3)) \\ G. C. Greubel, Aug 13 2019
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Sage
b=binomial; [1]+[b(2*n, n)-b(2*(n-1), n-1) +2*n -3 for n in (2..25)] # G. C. Greubel, Aug 13 2019
Formula
a(0)=1 for n>0 a(n)=binomial(2*n, n)-binomial(2*n-2, n-1)+2*n-3. - Benoit Cloitre, Sep 10 2003
G.f.: ((1-x)^3 - (1-2*x-x^3)*sqrt(1-4*x))/((1-x)^2*sqrt(1-4*x)). - G. C. Greubel, Aug 13 2019
Extensions
More terms from Benoit Cloitre, Sep 10 2003