A016864
a(n) = (5*n + 1)^4.
Original entry on oeis.org
1, 1296, 14641, 65536, 194481, 456976, 923521, 1679616, 2825761, 4477456, 6765201, 9834496, 13845841, 18974736, 25411681, 33362176, 43046721, 54700816, 68574961, 84934656, 104060401, 126247696
Offset: 0
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Table[(5*n + 1)^4, {n, 0, 25}] (* Amiram Eldar, Oct 02 2020*)
LinearRecurrence[{5,-10,10,-5,1},{1,1296,14641,65536,194481},30] (* Harvey P. Dale, Jul 22 2021 *)
A016898
a(n) = (5*n + 4)^2.
Original entry on oeis.org
16, 81, 196, 361, 576, 841, 1156, 1521, 1936, 2401, 2916, 3481, 4096, 4761, 5476, 6241, 7056, 7921, 8836, 9801, 10816, 11881, 12996, 14161, 15376, 16641, 17956, 19321, 20736, 22201, 23716, 25281, 26896, 28561, 30276, 32041, 33856, 35721, 37636, 39601, 41616
Offset: 0
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[(5*n+4)^2: n in [0..70]]; // Vincenzo Librandi, May 02 2011
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Table[(5*n + 4)^2, {n, 0, 25}] (* Amiram Eldar, Oct 02 2020 *)
LinearRecurrence[{3,-3,1},{16,81,196},50] (* Harvey P. Dale, Jul 30 2023 *)
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Vec((16 + 33*x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Mar 30 2017
A347533
Array A(n,k) where A(n,0) = n and A(n,k) = (k*n + 1)^2 - A(n,k-1), n > 0, read by ascending antidiagonals.
Original entry on oeis.org
1, 2, 3, 3, 7, 6, 4, 13, 18, 10, 5, 21, 36, 31, 15, 6, 31, 60, 64, 50, 21, 7, 43, 90, 109, 105, 71, 28, 8, 57, 126, 166, 180, 151, 98, 36, 9, 73, 168, 235, 275, 261, 210, 127, 45, 10, 91, 216, 316, 390, 401, 364, 274, 162, 55, 11, 111, 270, 409, 525, 571, 560, 477, 351, 199, 66
Offset: 1
Array, A(n, k), begins:
1 3 6 10 15 21 28 36 45 ... A000217;
2 7 18 31 50 71 98 127 162 ... A195605;
3 13 36 64 105 151 210 274 351 ...
4 21 60 109 180 261 364 477 612 ...
5 31 90 166 275 401 560 736 945 ...
6 43 126 235 390 571 798 1051 1350 ...
7 57 168 316 525 771 1078 1422 1827 ...
8 73 216 409 680 1001 1400 1849 2376 ...
9 91 270 514 855 1261 1764 2332 2997 ...
Antidiagonals, T(n, k), begin as:
1;
2, 3;
3, 7, 6;
4, 13, 18, 10;
5, 21, 36, 31, 15;
6, 31, 60, 64, 50, 21;
7, 43, 90, 109, 105, 71, 28;
8, 57, 126, 166, 180, 151, 98, 36;
9, 73, 168, 235, 275, 261, 210, 127, 45;
10, 91, 216, 316, 390, 401, 364, 274, 162, 55;
Family of sequences (k*n + 1)^2:
A016754 (k=2),
A016778 (k=3),
A016814 (k=4),
A016862 (k=5),
A016922 (k=6),
A016994 (k=7),
A017078 (k=8),
A017174 (k=9),
A017282 (k=10),
A017402 (k=11),
A017534 (k=12),
A134934 (k=14).
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A347533:= func< n,k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;
[A347533(n,k): k in [0..n-1], n in [1..13]]; // G. C. Greubel, Dec 25 2022
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A[n_, 0]:= n; A[n_, k_]:= (k*n+1)^2 -A[n,k-1]; Table[Function[n, A[n, k]][m-k+1], {m,0,10}, {k,0,m}]//Flatten (* Michael De Vlieger, Oct 27 2021 *)
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def A347533(n,k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))
flatten([[A347533(n,k) for k in range(n)] for n in range(1,14)]) # G. C. Greubel, Dec 25 2022
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