A026907 -id:A026907 - cp's OEIS frontend

A026918 Sum{T(n-k,k)}, 0<=k<=[ n/2 ], T given by A026907.

1, 13, 41, 90, 185, 335, 601, 1020, 1729, 2857, 4721, 7710, 12593, 20459, 33241, 53880, 87337, 141421, 229001, 370650, 599921, 970823, 1571041, 2542140, 4113505, 6655945, 10769801, 17426070, 28196249, 45622667

Offset: 0

Clark Kimberling

nonn

Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-3,1,1).

LinearRecurrence[{1,3,-2,-3,1,1},{1,13,41,90,185,335},30] (* Harvey P. Dale, Jan 08 2015 *)

8*Fibonacci(n+5) + (3/4)*[(n-1)(-1)^n - 17(n+3)]. - Ralf Stephan, May 13 2004

G.f. ( -1-12*x-25*x^2-12*x^3-x^4 ) / ( (x^2+x-1)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, May 22 2013

A026919 Uniquification of A026907.

Original entry on oeis.org

1, 13, 28, 44, 46, 67, 90, 91, 118, 148, 154, 181, 198, 217, 239, 256, 298, 343, 348, 370, 391, 442, 484, 496, 553, 613, 627, 650, 676, 742, 758, 811, 849, 883, 958, 993, 1036, 1084, 1117, 1201, 1288, 1358, 1378, 1403, 1471, 1495

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A026907.

A026919= Table[Binomial[n,k] +3*Binomial[n+4,k+2] -18, {n,0,100}, {k,0,n}]//Flatten// Union;
Table[A026919[[n]], {n,100}] (* G. C. Greubel, Aug 22 2025 *)

def A026907(n,k): return binomial(n,k) +3*binomial(n+4,k+2) -18
A026919= sorted(set(flatten([[A026907(n,k) for k in range(n+1)] for n in range(103)]) ))
print([A026919[n] for n in range(100)]) # G. C. Greubel, Aug 22 2025

A027170 Triangular array T read by rows (4-diamondization of Pascal's triangle). Step 1: t(n,k) = C(n+2,k+1) + C(n+1,k) + C(n+1,k+1) + C(n,k). Step 2: T(n,k) = t(n,k) - t(0,0) + 1. Domain: 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 3, 3, 5, 10, 5, 7, 19, 19, 7, 9, 30, 42, 30, 9, 11, 43, 76, 76, 43, 11, 13, 58, 123, 156, 123, 58, 13, 15, 75, 185, 283, 283, 185, 75, 15, 17, 94, 264, 472, 570, 472, 264, 94, 17, 19, 115, 362, 740, 1046, 1046, 740, 362, 115, 19, 21, 138, 481, 1106, 1790, 2096, 1790, 1106, 481, 138, 21

Offset: 0

Clark Kimberling

			Triangle starts:
   1;
   3,  3;
   5, 10,  5;
   7, 19, 19,  7;
   9, 30, 42, 30,  9;
  11, 43, 76, 76, 43, 11;
  ...

Indranil Ghosh, Rows of n = 0..125 of triangle, flattened

Cf. A007318, A026907.

t[n_, k_]:= Binomial[n + 2, k + 1] + Binomial[n + 1, k] + Binomial[n + 1, k + 1] + Binomial[n ,k]; T[n_, k_] := t[n, k] - t[0, 0] + 1; Flatten[Table[T[n, k], {n, 0, 10},{k, 0, n}]] (* Indranil Ghosh, Mar 13 2017 *)

alias(C, binomial);
t(n,k) = C(n+2,k+1)+C(n+1,k)+C(n+1,k+1)+C(n,k);
T(n,k) = t(n,k)-t(0,0)+1;
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print());
\\ Michel Marcus, Mar 13 2017

cp's OEIS Frontend

A026918 Sum{T(n-k,k)}, 0<=k<=[ n/2 ], T given by A026907.

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A026919 Uniquification of A026907.

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A027170 Triangular array T read by rows (4-diamondization of Pascal's triangle). Step 1: t(n,k) = C(n+2,k+1) + C(n+1,k) + C(n+1,k+1) + C(n,k). Step 2: T(n,k) = t(n,k) - t(0,0) + 1. Domain: 0 <= k <= n, n >= 0.

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